Quillen–Suslin theorem

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For the now-solved conjecture on Galois representations, see Serre conjecture (number theory).

The Quillen–Suslin theorem, also known as Serre's problem or Serre's conjecture, is a theorem in commutative algebra about the relationship between free modules and projective modules over polynomial rings. Every free module over a ring is projective, but most rings admit projective modules that are not free. Serre asked whether the converse holds in certain situations.

Geometrically, finitely generated free modules correspond to trivial vector bundles and finitely generated projective modules to more general vector bundles. Affine space is topologically contractible, so it admits no non-trivial topological vector bundles. A simple argument using the exponential exact sequence and the d-bar Poincaré lemma shows that it also admits no non-trivial holomorphic vector bundles. Jean-Pierre Serre, in his 1955 paper "Faisceaux algébriques cohérents", remarked that the equivalent question was not known for algebraic vector bundles: "It is not known if there exist projective A-modules of finite type which are not free."^[1] Here A is a polynomial ring over a field, that is, A = k[x₁, ..., x_n].

To Serre's dismay, this problem quickly became known as Serre's conjecture. (Serre wrote, "I objected as often as I could [to the name]."^[2]) The statement is not immediately obvious from the topological and holomorphic cases, because these cases only guarantee that there is a continuous or holomorphic trivialization, not an algebraic trivialization. Instead, the problem turns out to be extremely difficult. Serre made some progress towards a solution in 1957 when he proved that every finitely generated projective module over a polynomial ring over a field was stably free, meaning that after forming its direct sum with a free module, it became free. The problem remained open until 1976, when Daniel Quillen and Andrei Suslin, independently proved that the answer was affirmative. Quillen was awarded the Fields Medal in 1978 in part for his proof of the Serre conjecture. Leonid Vaseršteĭn later gave a simpler and much shorter proof of the theorem which can be found in Serge Lang's Algebra.

[edit] References

• Serre, Jean-Pierre (1955), “Faisceaux algébriques cohérents”, Annals of Mathematics. Second Series. 61: 197-278, <http://links.jstor.org/sici?sici=0003-486X%28195503%292%3A61%3A2%3C197%3AFAC%3E2.0.CO%3B2-C> 
• Serre, Jean-Pierre (1958), “Modules projectifs et espaces fibrés à fibre vectorielle”, Séminaire P. Dubreil, M.-L. Dubreil-Jacotin et C. Pisot, 1957/58, Fasc. 2, Exposé 23 
• Quillen, Daniel (1976), “Projective modules over polynomial rings”, Inventiones Mathematicae 36: 167-171, DOI 10.1007/BF01390008 
• Suslin, Andrei A. (1976), “Projective modules over polynomial rings are free”, Dokl. Akad. Nauk SSSR 229 (5): 1063-1066 . Translated in Suslin, Andrei A. (1976), “Projective modules over polynomial rings are free”, Soviet Math. Dokl. 17 (4): 1160-64 
• Lang, Serge (2002), Algebra (Revised third ed.), Graduate Texts in Mathematics, 211, Springer Science+Business Media, ISBN 0-387-95385-X 

An account of this topic is provided by:

• Lam, T. Y. (2006), Serre's problem on projective modules, Berlin; New York: Springer Science+Business Media, pp. 300pp., ISBN 978-3540233176